Questions about .apk files

So I understand how to install apps from their .apk files using adb (install filepath), and that's all very good and handy if the .apk is on my computer.

But if I download an .apk straight to my phone because I'm on the go, and I install it on my phone, can I then delete the .apk file if it's unnecessary? Or will the app cease to function if I delete the original installation .apk because the app is always run directly from this file?

Also, is this process of installing apps from their .apk installation file is what people refer to as "sideloading?" Because I didn't do the instructions I see everywhere to "enable sideloading on at&t" but I've installed apps via their .apk file and adb..

Sideloading is the installing of apks directly through your phone. You don't use the market to download and install. In sideloading I generally use a file manager like Astro or Root Explorer.

If you download the apk (from the browser not the market) and installed it straight to your phone, then you can remove the apk file it won't need it. That file is generally found in the downloads folder on your SD card or phone. Using adb push to your phone, you're generally pushing the apk to system folder, or it could be just the data folder. Sideloading is generally easier, especially for someone like you and I, since we're always on the go, that's if we're not using the market.